Some applications of the parabolic maximum principles to nonlinear parabolic differential equations

In this paper, we develop some estimates for the time derivative and the gradient of the solution of a nonlinear parabolic differential equation with various boundary conditions assigned to it. The method employed involves the derivation of a parabolic differential equation (or inequality) for a functional defined on the solution of the initial-boundary value problem, and then applying the parabolic maximum principles to obtain a bound for the functional.